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Let G be a cyclic group. There's a theorem which states that if |G| is a prime, then every non-identity member of G is a generator. What about a cyclic group whose order is not prime: Is there such a group whose every non-identity member is a generator? Are there other necessary/sufficient conditions regarding groups whose every non-identity member is a generator? (Beyond primality of |G|.) |
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To answer your first question: no, a cyclic group whose order is not prime must contain non-identity (thanks Zev!) elements that are not generators. Let $G$ be a cyclic group, let $g$ be any generator of $G$, and let $n$ be the order of $G$. Then for any $d$ that divides $n$, the subgroup generated by $g^d$ is not all of $G$ (this subgroup has $n/d$ elements, but $G$ has $n$ elements). To answer your second question: for every non-identity element of $G$ to be a generator, $G$ must be a cyclic group with prime order. If $G$ weren't a cyclic group, then $G$ wouldn't have any generators at all (the definition of "cyclic group" is "a group that can be generated by a single element"), and the answer to your first question shows that the order of $G$ must be prime. |
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